Inhomogeneous Partial Differential Equations Research Articles

The High-Order Perturbation Approximate Solution of the Finite Ultrasonic Wave

To the nonlinear acoustic wave equation, the general used second harmonic solution is not accurate enough because all perturbated expansion equations higher than the second order are ignored during the equation solving process. The purpose of this paper is to obtain a more accurate solution, i.e., the high-order perturbation approximate solution. Firstly, the nonlinear acoustic wave equation is expanded into many inhomogeneous partial differential equations. the low-order harmonic solutions are obtained manually, then we formulate the form of the high-order harmonic solutions according to the properties of the low-order harmonic solutions. Using symbol calculation tool, we finally obtained higher up to the 14th order perturbation special solutions. Odd order solutions contain only odd order harmonics, and even order solutions contain only even order harmonics. The high-order perturbation solution of the second harmonic is finally achieved by summing up all of the second harmonic solution parts. The simulation results show that the relative amplitude (A 2/A 1) of the second harmonic increases and then decreases with the propagation distance, which is in agreement with experimental results. The high-order perturbation approximated solution can compensate for the theory deficiency and can be used to measure the nonlinear parameter with a good precision.

Some new applications of β-Laplace integral transform

The β-Laplace integral transform(β-LIT) has been introduced recently by Gaur et. al. which is a generalization of Laplace transform. This new β-LIT shows some special properties over Laplace integral transform. In the current paper, we investigate the applicability of the β-LIT technique to solve partial differential equation such as initial value problem of first order and second order PDE, wave equation, Diffusion equation, Potential and current equation, Heat conduction equation, Axisymmetric Heat conduction equation, Inhomogeneous partial differential equation, inhomogeneous wave equation, Stroke and Rayleigh problem in fluid dynamics and illustrate by some examples.

Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of $${\mathbb {C}}^{n+1}$$ . We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any $$s\ge s_c$$ , where $$s_c$$ is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case $$s<s_c$$ , we show that the solution is generically $$s_c$$ -Gevrey while the inhomogeneity is s-Gevrey, and we give an explicit example in which the solution is $$s'$$ -Gevrey for no $$s'<s_c$$ .

A diffuse interface model of reactive-fluids and solid-dynamics

This article presents a multi-physics methodology for the numerical simulation of physical systems that involve the non-linear interaction of multi-phase reactive fluids and elastoplastic solids, inducing high strain-rates and high deformations. Each state of matter is governed by a single system of non-linear, inhomogeneous partial differential equations, which are solved simultaneously on the same computational grid, and do not require special treatment of immersed boundaries. To this end, the governing equations for solid and reactive multiphase fluid mechanics are written in the same mathematical form and are discretised on a regular Cartesian mesh. All phase and material boundaries are treated as diffuse interfaces. An interface-steepening technique is employed at material boundaries to keep interfaces sharp whilst maintaining the conservation properties of the system. These algorithms are implemented in a highly-parallelised hierarchical adaptive mesh refinement platform, and are verified and validated using numerical and experimental benchmarks. Results indicate very good agreement with experiment and an improvement of numerical performance compared to certain existing Eulerian methods, without loss of conservation.

Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients

In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined by the linear part of the equation. These conditions generalize the ones given by the author for the linear case (Remy in J Dyn Control Syst 22(4):693–711, 2016; J Dyn Control Syst 23(4):853–878, 2017) and for the semilinear heat equation (Remy in J Math Anal Appl 494(2):124619, 2021). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.

Exact solutions of the Navier–Stokes equations for describing an isobaric one-directional vertical vortex flow of a fluid

The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolic-type equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).

The diffusion problem

The study of diffusion processes is relevant in various fields: chemistry, physics, metallurgy, medicine, and others. The paper considers and solves the problem of diffusion of charged particles under the action of an electromagnetic field in a semi-infinite thin tube. The mathematical formulation is a boundary value problem for an inhomogeneous partial differential equation of the parabolic type. To obtain an accurate analytical solution, the operator method is used. The solution is initially obtained in the form of an image, and then the solution of the original problem is obtained using the inverse Laplace transform. The solution of the test example is obtained, a graphical image of the desired function is constructed in the MatLab software package.

Numerical Solution of the Vibration Suppression Problem for a Moving Web

Mechanical processes occurring in paper production are modeled. In a papermaking machine, paper moves in the form of a thin sheet. The characteristic thickness of a sheet varies from 0.1 mm (office paper) to 1 mm (cardboard). Every papermaking machine contains open segments where a paper web passes without mechanical support in its motion from one roller to another. On such segments, the web may lose stability, performing transverse vibrations, and, as a result, might tear. The possibility of reducing these vibrations with the help of various control actuators is explored. The transverse vibrations of a moving web with nonzero bending stiffness are modeled by a fourth-order inhomogeneous partial differential equation. The action of control actuators is modeled a function on the right-hand side of the equation. The vibration amplitude is assumed to be identical across the moving web. The vibration suppression problem is reduced to the minimization of a multivariable function. The solution of the problem splits into two stages: the solution of an initial-boundary value problem with a given control and the minimization of a multivariable function. A numerical method is proposed for solving the initial-boundary value problem. The fourth-order differential equation is reduced to a system of two second-order differential ones. The latter are simplified by changing the sought functions. The resulting equations are approximated by a finite-difference scheme, which is proved to be absolutely stable. This scheme is solved using block Gaussian elimination. The multivariable function is minimized by applying the Hooke–Jeeves method. Examples of computations are given for actuators of three types, namely, point actuators, actuators acting on a web segment, and actuators acting along the entire web.

Solving the boundary problem by the method of green's function for the simple differential equation of the second order linear

Green functions are used mainly to solve certain types of linear inhomogeneous partial differential equations, although homogeneous partial differential equations can also be solved using this approach. In this article is discussed about how to solve or how to find solution of differential equations using Green's function. Furthermore, after theory, some examples are given to explain.

New iterative approach for the solutions of fractional order inhomogeneous partial differential equations

In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.

Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

Based on the method of dynamic programming, this paper uses analysis methods governed by the nonlinear and inhomogeneous partial differential equation to study modern portfolio management problems with stochastic volatility, incomplete markets, limited investment scope, and constant relative risk aversion (CRRA). In this paper, a three-level Crank–Nicolson finite difference scheme is used to determine numerical solutions under this general setting. One of the main contributions of this paper is to apply this three-level technology to solve the portfolio selection problem. In addition, we have used a technique to deal with the nonlinear term, which is another novelty in performing the Crank–Nicolson algorithm. The Crank–Nicolson algorithm has also been extended to third-order accuracy by performing Richardson’s extrapolation. The accuracy of the proposed algorithm is much higher than the traditional finite difference method. Lastly, experiments are conducted to show the performance of the proposed algorithm.

Asymptotics of the Solution to the Boundary-Value Problems with Non Smooth Coefficient

The first boundary-value problem for a linear inhomogeneous second-order partial differential equation of elliptic type with two independent variables in a ring is investigated in the case when the potential is a non smooth function. Using the modified method of boundary functions, a uniform asymptotic expansion of the solution of the Dirichlet problem is constructed. The maximum principle is used to obtain an estimate for the remainder term of the asymptotic expansion. The resulting series is asymptotic in the sense of Erdey.

Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model

AbstractThis work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

Green function solution of generalised boundary value problems

We construct an expression for the Green function of a differential operator satisfying nonlocal, homogeneous boundary conditions starting from the fundamental solution of the differential operator. This also provides the solution to the boundary value problem of an inhomogeneous partial differential equation with inhomogeneous, nonlocal boundary conditions. The construction applies for a broad class of linear partial differential equations and linear boundary conditions.

Solution of model boundary value problems on oscillations of mechanical systems with moving boundaries by the duhamel method

The Kantorovich – Galerkin method in conjunction with the Duhamel method is considered as applied to solving problems describing the oscillations of visco–elastic objects with conditions on moving boundaries. The mathematical formulation of the problem includes an inhomogeneous partial differential equation with respect to the desired displacement function, inhomogeneous boundary and initial conditions. By introducing a new function into the problem, the boundary and initial conditions are reduced to homogeneous. The solution is made in dimensionless variables with an accuracy of the second order of smallness with respect to small parameters characterizing the velocity of the boundary and viscoelasticity. Using the Kantorovich – Galerkin method and the Duhamel method, an approximate solution of the problem of forced longitudinal oscillations of a viscoelastic rod of variable length is found.

Exact Analytic Solution and Investigation of the Guyer-Krumhansl Heat Equation

The heat propagation in a Guyer-Krumhansl one-dimensional thermal conductivity model is studied. The exact analytic solutions of a one-dimensional equation of the Guyer-Krumhansl type are obtained and investigated by the operator method, taking into account the medium motion, and also a harmonic solution for a system of third-order inho-mogeneous partial differential equations is obtained. The evolution is investigated for several types of initial distributions with periodic, power-law, exponential-power-law coordinate dependence and for the distribution of the Dirac delta function. The exact solutions thus obtained can simulate, in particular, the propagation of heat pulses of diverse forms under real experimental conditions, such as an isolated spatial heat wave, a laser pulse, and a periodic heating. The results are applied to simulate the propagation of heat in ultrathin films. The problem of heat propagation in small-size systems like nanofibres and nanofilms is solved analytically, taking into account ballistic effects; the maximum principle and the positivity of solutions in the context of heat conduction are discussed.

An infinite sequence of conserved quantities for the cubic Gross–Pitaevskii hierarchy on ℝ

We consider the cubic Gross–Pitaevskii (GP) hierarchy on R \mathbb {R} , which is an infinite hierarchy of coupled linear inhomogeneous partial differential equations appearing in the derivation of the cubic nonlinear Schrödinger equation from quantum many-particle systems. In this work, we identify an infinite sequence of operators which generate infinitely many conserved quantities for solutions of the GP hierarchy.

Partition of unity extension of functions on complex domains

We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in 2D. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to 10−14.

Wave Excitation in a Dissipative Medium with a Double Quadratically-Modular Nonlinearity: a Generalization of the Inhomogeneous Burgers Equation

Solutions of a forced (inhomogeneous) partial differential equation of the second order with two types of nonlinearity: power (quadratic) and nonanalytic (modular) are found. Equations containing each of these nonlinearities separately were studied earlier. A natural continuation of these studies is the development of the theory of wave phenomena in a medium with a double nonlinearity, which have recently been observed in experiments. Here solutions describing the profiles of intense waves are derived. Shapes of freely propagating stationary perturbations in the form of shock waves with a finite front width are found. The profiles of forced waves excited by external sources are calculated.

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.